Fractional quantum ferroelectricity

For an ordinary ferroelectric, the magnitude of the spontaneous electric polarization is at least one order of magnitude smaller than that resulting from the ionic displacement of the lattice vectors, and the direction of the spontaneous electric polarization is determined by the point group of the ferroelectric. Here, we introduce a new class of ferroelectricity termed Fractional Quantum Ferroelectricity. Unlike ordinary ferroelectrics, the polarization of Fractional Quantum Ferroelectricity arises from substantial atomic displacements that are comparable to lattice constants. Applying group theory analysis, we identify 28 potential point groups that can realize Fractional Quantum Ferroelectricity, including both polar and non-polar groups. The direction of polarization in Fractional Quantum Ferroelectricity is found to always contradict with the symmetry of the “polar” phase, which violates Neumann’s principle, challenging conventional symmetry-based knowledge. Through the Fractional Quantum Ferroelectricity theory and density functional calculations, we not only explain the puzzling experimentally observed in-plane polarization of monolayer α-In2Se3, but also predict polarization in a cubic compound of AgBr. Our findings unveil a new realm of ferroelectric behavior, expanding the understanding and application of these materials beyond the limits of traditional ferroelectrics.


G F -P L Pairs of FQFE
A structure L1 (belong to space group GL) that has FQFE can be divided into two parts, i.e.F and M. The space group of F, GF, is the supergroup of GL.The Wyckoff position of each atom M i in GF should be one listed in Tab.S1.In this section, we first constructed the ferroelectric phase L2 from a given L1 phase of monolayer Sc2CO2.Then we discuss the FQFE contributed by several atoms.

The Wyckoff Position Tables of P-3m1 and Fm-3m
In the case of the monolayer α-In2Se3 and Sc2CO2, GF is P-3m1 and the Wyckoff position table is shown in Table S2.For AgBr, GF is Fm-3m and the Wyckoff position table is shown in Tab.S3.

Ferroelasticity of non-polar FQFE
In the ferroelectric phase transition of a system with a square H phase (see Fig. 1(c) in the main text), the low symmetry phases L1 and L2 do not have the four-fold rotational symmetry.Thus, there should be lattice distortion in the two phases (see Fig. S3) and in this case the FQFE can be ferroelastic.

The thermal-stability analysis of FQFE materials
We perform ab initio molecular dynamics (AIMD) simulations for the FQFE systems of monolayer In2Se3 and bulk AgBr, for both the FE state and the intermediate state.As shown in Fig. S4, both example FQFE systems exhibit good thermal-stability at room-temperature.

FQFE in organic-inorganic materials
The structure of NH 4 Br is constructed by replacing the Ag atom of AgBr by NH4 + .
The fractional polarization is similar to that in AgBr.This can also be considered as an instance of FQFE with multi-atom movements.

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Table.S1.All the possible G F -P L pairs that may exhibit FQFE with one mobile M atom.The site symmetry group of the Wyckoff position (letter) in G F is P L , the point group of the low symmetry phase. is the denominator of the fraction in in the conventional cell. is the polarization quantum.

Figure S2 .
Figure S2.Dynamical and thermal stability of AgBr.(a) Phonon spectrum along the highsymmetry path Γ → X → S → Y → Γ → Z → U → R → T → Z of AgBr for F-43m phase.(b) Ab initio molecular dynamics simulations of AgBr for F-43m phase at 300 K.The inset shows the corresponding structure after 5 ps of simulation.

Figure S3 .
Figure S3.Ferroelasticity of non-polar FQFE, corresponding to the phase transition of Fig. 1(c) in the main text.

Fig. S4
Fig. S4 Thermal stability of FQFE materials.(a) and (b) are the ab initio molecular dynamics (AIMD) simulations of AgBr for F-43m and Fm-3m phase (L and H phase) at 300 K. (c) and (d) are the AIMD simulations of monolayer α-In 2 Se 3 for FE and fcc' phase at 300 K.The inset shows the corresponding structure after 5 ps of simulation.

Fig
Fig. S5 Illustration of realizing FQFE in molecular material of NH 4 Br, where the NH 4 + takes the position of Ag in AgBr.

Fig. S7
Fig. S7 The evolution of the polarization for monolayer α-In 2 Se 3 in the primitive cell along the paths in Fig. S6: (a) PE phase (b) fcc' phase (c) H phase (d) fcc'' phase. is the polarization quantum along the [110] direction.The polarization difference ∆P=P 1 -P 2 are 4.74 eÅ, 4.74 eÅ, -9.48 eÅ and -9.48 eÅ per unit cell for (a), (b), (c), (d), respectively.The polarization of the gray points in panel (a) is determined by fitting other structures in this path due to their metallic behavior.

Table .
S2. Wyckoff position table1of No.164 space group, P-3m1.In α-In 2 Se 3 , M is the middle Se atom and its Wyckoff position in G F =P-3m1 is 2d.